New Exact Solutions of Kolmogorov Petrovskii Piskunov Equation, Fitzhugh Nagumo Equation, and Newell- Whitehead Equation

Department of Mathematics, Huzhou University, Huzhou 313000, China Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, China Department of Mathematics, COMSATS University Islamabad, Park Road, Chak Shahzad, Islamabad, Pakistan Department of Mathematics, Cankaya University, Ankara, Turkey Institute of Space Sciences, Magurele-Bucharest, Romania Department of Mathematics, Riphah International University, Sector I-14, Islamabad, Pakistan Faculty of Engineering Technology, Amol University of Special Modern Technologies, Amol, Iran

Feng presented an effective technique to obtain travelling wave solutions of NPDEs, known as the FIM method [20][21][22]. FIM is based on the ring theory and commutative algebra. FIM provides first integral of explicit form having polynomials as coefficients by applying the division theorem. Contrary to other methods, the benefits of FIM are to produce exact and explicit solutions without complicated and lengthy calculations [23][24][25]. Despite several advantages, FIM can only be applied to integrable PDEs.
The focus of this paper is to find the exact solutions of conformable biological models. It includes KPP and its derived models, namely, FHN and NW. KPP is a general form of equation and we can obtain different equations from KPP, for example, FHN, NW, and Cahn Allen. The considered models are significant in biology. KPP describes the genetic model for spread of dominant gene through population. The graphical solutions of the KPP equation can be used for diallel analysis as diallel analysis requires graphical solutions of genes. In diallel analysis, graphical representation of genes is required and some further calculations enable researchers to have point estimation of recessive genes and dominant genes instead of providing an interval of estimation [26]. The FHN equation is used in the study of intercellular trigger waves. Trigger waves are pulses and oscillatory waves; these waves switch from one stable steady state to another [27]. Similarly, the NW equation is applied for Faraday instability, chemical reactions, Rayleigh-Benard convection, and biological systems [28]. Various techniques have been established for solving the KPP equation, for instance, the discrimination algorithm [29], the homotopy perturbation technique [30], the differential transform method [31], the (G′/G)-expansion method [32], and the homotopy analysis method [33]. Generally, the solutions of KPP equations are based on series solutions or numerical solutions. In this work, an effective technique named as FIM was adopted to acquire the exact solutions of KPP, FHN, and NW equations. The work is novel as the exact solutions of considered models using FIM are not presented before in the literature.
This paper consists of the following sections. Conformable derivative is described in Section 2; the proposed technique FIM is discussed in Section 3; the solutions of conformable KPP, FHN, and NW equations are presented in Section 4, and Section 5 contains summary and further recommendations.
The conformable integral for function h is given as where p ≥ 0 and β ∈ ð0, 1.
The conformable derivative of any differential function at origin is zero; despite this flaw, several studies have been made on conformable derivative, as it explains higher order integration, sequential differentiation and integration, connection of differentiation and integration, property of linearity, derivative of constant function, quotient and product rule, chain rule, and power rule [34,[36][37][38][39]. Consequently, many researchers are working on the applicability of conformable derivative for real-world problems, such as Jacobi elliptical function expansion method used to solve conformable Boussinesq and combined Kdv-mKdv equation [40], conformable space-time fractional (2+1) dimensional dispersive long wave equation [41], conformable heat equation [42], and conformable perturbed nonlinear Schrodinger equation [43].

Methodology
Here, the methodology of FIM is presented.
Step 1. Conformable PDE is given as follows: Step 2. Now using the following transformation Specifically in case of conformable derivative, the next transformation is applied as The transformation defined in equation (5) will convert conformable PDE in nonlinear ODE.

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where U ′ ðYÞ = dUðYÞ/dY and transformed variable is denoted by Y.
Step 3. We will take other independent variables as As a result, FIM will provide a system of ODEs (nonlinear) as Step 4. We attain general solutions after integrating equation (8). There is no precise or sound technique to obtain first integrals in case of plane independent (autonomous) system, so it is difficult to get even one first integral. To determine the first integral, the division theorem is utilized. Hence, a first integral is derived (cf. equation (8)) with the help of the division theorem. In this way, nonlinear ordinary differential equations (ODEs) can be reduced into a first-order ODE system (integrable) through the division theorem. Afterwards, solving the obtained system (cf. equation (8)), the exact solutions can be acquired.
The theorem for complex domain ℂ and two variables is given as Division Theorem. Consider polynomials Gðz, yÞ and Rðz, yÞ, in complex domain ℂ, where GðzyÞ is irreducible. If at all zero points of Rðz, yÞ, Gðz, yÞ vanishes, then another polynomial Hðz, yÞ exists in ℂðz, yÞ and the following equality holds:

Implementation of FIM: Conformable KPP Equation and Its Derived Equations
The exact solutions of KPP, FHN, and NW equations are presented in this section.

Conformable Space-Time Fractional KPP Equation.
Andrey Kolmogorov, Ivan Petrovsky, and Nikolai Piskunov proposed a nonlinear PDE called the Kolmogorov Petrovsky Piskunov (KPP) equation to describe the genetic model for spread of dominant gene through population. Later, the KPP equation is applied in different natural sciences like in physics as combustion, in biology as propagation of nerve impulses, in chemical kinetics as propagation of concentration waves, and in plasma as evolution of set of duffing oscillators.
Consider conformable space-time fractional KPP equation defined as [32,44] where μ, η, and δ are constants and β ∈ ð0, 1Þ. First, we use conformable derivative with the following transformation: where the transformation variable is Y. The transformation represented in equation (11) will provide the following conversions: Here, m and p are constants. Then, we get ODE by using equation (12) in equation (10): Now, we acquire a 2D system from equation (7) as Afterwards, the division theorem will provide first integral. According to FIM, Z and Y are supposed to be nontrivial solutions of the system given (cf. equations (14) and (15)). Now, the division theorem provides us an irreducible polynomial RðZ, YÞ = ∑ n r=0 a r ðZÞY r in ℂ½Z, Y given as where a r ðZÞ ≠ 0 and r = 0, 1, ⋯, n. Now, we have a polynomial of form wðZÞ + qðZÞY in ℂ½Z, Y such that Using n = 1 in equation (17) and equating coefficients for Y r ðr = 0, 1Þ, then we have following equations: 3 Advances in Mathematical Physics Here, a r ðZÞ are polynomials in Z. As equation (18) shows a 1 ðZÞ has constant nature, hence qðZÞ = 0 and we can take a 1 ðZÞ = 1. We conclude that deg ðwðZÞÞ can only be 0 or 1 by using a 1 ðZÞ and qðZÞ in equations (19) and (20) and after balancing the functions wðZÞ and a 0 ðZÞ degrees. Now, we can take wðZÞ = A 1 Z + A 0 ; therefore, equation (19) takes the following form: where A 2 is an integrating constant. Afterwards, the substitutions of values of a 0 ðZÞ, wðZÞ in equation (20) provide a system of nonlinear algebraic equations by equating the power of Z. Thus, as a result, we obtain various values of constants given as follows.
Case 1. The following constants are acquired as follows: Substituting equations (21) and (22) into equation (16), we get Substitution of equation (23) into equation (14) provides the first solution of conformable fractional KPP equation.

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Substitution of equation (32) into equation (14) provides the fourth solution of conformable fractional KPP equation.
The solutions u 1 , u 2 , u 3 , u 4 , u 5 , u 6 are presented in Figure 1. For larger values of β, the solutions attain more height which is depicted in Figure 2. Figure 3 shows the graphical presentation in 2D plot of genes for the KPP equation. The graphical solutions of the KPP equation can be used in diallel analysis as observed in [26].

Conformable Time-Fractional FHN Equation.
Richard Fitzhugh proposed a model for transmission of impulses in nerve axon in 1961. Nagumo et al. made the identical circuit in succeeding years and presented the model of an excitable system. FHN is derived from KPP on substituting μ = ξ, η = −ðξ + 1Þ [45,46].
Consider conformable time-space fractional FHN equation as where β ∈ ð0, 1Þ and ξ ∈ ð0,0:5. First, we use the conformable derivative with the following transformation: where the transformation variable is Y. The transformation represented in equation (41) will provide the following conversions: Here, p is a constant. Then, we get an ODE by using equation (42) in equation (40): Now, we obtain a 2D system from equation (7) as Afterwards, the division theorem will give first integrals. According to FIM, Z and Y are supposed to be nontrivial solutions of the system given (cf. equations (44) and (45)). Now, the division theorem provides us an irreducible polynomial RðZ, YÞ = ∑ n r=0 a r ðZÞY r in ℂ½Z, Y as Using n = 1 in equation (47) and equating coefficients of Y r ðr = 0, 1Þ, then we have the following equations: Here, a r ðZÞ are polynomials in Z. As equation (48) shows a 1 ðZÞ has constant nature, thus qðZÞ = 0 and we can take a 1 ðZÞ = 1. We conclude that deg ðwðZÞÞ can only be 0 or 1 by using a 1 ðZÞ and qðZÞ in equations (48) and (49) and after balancing the functions wðZÞ and a 0 ðZÞ degrees. Now, we can take wðZÞ = A 1 Z + A 0 ; therefore, equation (49) takes the following form: where A 2 is an integrating constant. Afterwards, the substitutions of the values of a 0 ðZÞ, wðZÞ in equation (50) provide a system of nonlinear algebraic equations by equating the power of Z. Now, as a result, we have various constants given as follows.
Substitution of equation (53) into equation (44) provides the first solution of conformable fractional FHN equation.
Case 8. We get
Substitution of equation (56) into equation (44) provides the second solution of conformable fractional FHN equation.
Substitution of equation (59) into equation (44) provides the third solution of conformable fractional FHN equation.
The solutions u 7 , u 8 , u 9 , u 10 , u 11 , u 12 are presented in Figure 4. For smaller value of β, solutions attain more height which is depicted in Figure 5.
where p is a constant. Then, we get ODE by using equation (73) in equation (71).
Thus, we obtain a 2D system from equation (7) as Afterwards, the division theorem will give first integrals. According to FIM, Z and Y are supposed to be nontrivial solutions of the system given (cf. equations (75) and (76)). Hence, the division theorem provides us irreducible polynomial RðZ, YÞ = ∑ n r=0 a r ðZÞY r in ℂ½Z, Y given as where a r ðZÞ ≠ 0 and r = 0, 1, ⋯, n. Now, we have a polynomial of form wðZÞ + qðZÞY in ℂ½Z, Y such that Using n = 1 in equation (78) and equating coefficients of Y r ðr = 0, 1Þ, then we have following equations: Here, a r ðZÞ are polynomials in Z. As equation (79) shows a 1 ðZÞ has constant nature, thus qðZÞ = 0 and we can take a 1 ðZÞ = 1. We conclude that deg ðwðZÞÞ can only be 0 or 1 by using a 1 ðZÞ and qðZÞ in equations (80) and (81) and after balancing the functions wðZÞ and a 0 ðZÞ degrees. Now, we can take wðZÞ = A 1 Z + A 0 ; therefore, equation (80) takes the following form: where A 2 is an integrating constant. Afterwards, the substitutions of the values of a 0 ðZÞ, wðZÞ in equation (81) provide a system of nonlinear algebraic equations by equating coefficients of power of Z. Thus, we have various constants as given below.